This sounds to me like a lock-out. Basically, the control and/or ignition circuits shut down and do not allow furnace operation if potential safety problems are detected.
Are you on AC or DC right now? Low voltage could be a problem if you are on DC.
You can attempt to reset by turning the thermostat switch off and changing the set temperature to a low enough value so there is no demand for heat and waiting 10-15 seconds (maybe hard to do if its freezing out). Then turn the thermostat on and set the temperature to the desired level and the furnace should attempt to start again.
But, this doesn't really explain why it locked out in the first place. If not low voltage then it gets more complicated - the sail switch, high limit switch, control circuit board or loose wires could also be a problem.
Remember that lock-out is a safety feature. Its trying to keep something bad from happening. You might want to take this in if you are uncomfortable with any of this.
DW, 3 Kids and our Goldens
2012 Shamrock 233S
2008 Toyota Sequoia 5.7L 4WD